Numerical Analysis and Applications

, Volume 12, Issue 1, pp 1–14 | Cite as

Positive Periodic Solutions for a Class of Fourth-Order Nonlinear Differential Equations

  • N. BouteraaEmail author
  • S. Benaicha


In this paper, we obtain conditions of existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing explicit Green’s function and fixed point index theorem combining with operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish the theorem on the existence of positive solutions for a fourth-order boundary value problem with a variable parameter. Finally, we give an example to illustrate our results.


positive solution uniqueness and existence iterative sequence Green’s function 


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  1. 1.
    Fonda, A., Manasevich, R., and Zanolin, F., Subharmonic Solutions for Some Second-Order Differential Equations with Singularities, SIAM J. Math. An., 1993, vol. 24, no. 5, pp. 1294–1311.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cabada, A., Enguica, R., and Lopez-Somosa, L., Positive Solutions for Second-Order Boundary-Value Problems with Sign Changing Green’s Functions, Elect. J. Diff. Eq., 2017, no. 245, pp. 1–17.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Zhao, C.H., Chen, W., and Zhou, J.L., Periodic Solutions for a Class of Fourth-Order Nonlinear Differential Equations, Nonlin. An. TMA, 2010, vol. 72, nos. 3/4, pp. 1221–1226.Google Scholar
  4. 4.
    Guo, D. and Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Orlando, FL:Academic Press, 1988.zbMATHGoogle Scholar
  5. 5.
    Bonheure, D. and De Coster, C., Forced Singular Oscillators and the Method of Lower and Upper Solutions, Topol.Meth. Nonlin. An., 2003, vol. 22, no. 2, pp. 297–317.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Zeidler, E., Applied Functional Analysis, New York: Springer, 1995.zbMATHGoogle Scholar
  7. 7.
    Haddouchi, F. and Benaicha, S., Positive Solutions of a Nonlinear Three-Point Eigenvalue Problem with Integral Boundary Conditions, Romanian J. Math. Comp. Sci., 2015, vol. 5, iss. 2, pp. 202–213.Google Scholar
  8. 8.
    Han, G.D. and Xu, Z.B., Multiple Solutions of Some Nonlinear Fourth-Order Beam Equations, Nonlin. An. TMA, 2008, vol. 68, no. 12, pp. 3646–3656.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wang, H.Y., Positive Periodic Solutions of Singular Systems with a Parameter, J. Diff. Eq., 2010, vol. 249, no. 12, pp. 2986–3002.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rachunkova, I., Tvrdy, M., and Vrkoc, I., Existence of Nonnegative and Nonpositive Solutions for Second-Order Periodic Boundary Value Problems, J. Diff. Eq., 2001, vol. 176, no. 2, pp. 445–469.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sun, J. and Liu, Y., Multiple Positive Solutions of Singular Third-Order Periodic Boundary Value Problem, Acta Math. Scientia, Ser. B, 2005, vol. 25, no. 1, pp. 81–88.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chu, J.F., Torres, P.J., and Zhang, M.R., Periodic Solutions of Second-Order Non-Autonomous Singular Dynamical Systems, J. Diff. Eq., 2007, vol. 239, no. 1, pp. 196–212.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Xia, J. and Wang, Z.H., Existence and Multiplicity of Periodic Solutions for the Duffing Equation with Singularity, Proc. R. Soc. Edinb., Sect. A, 2007, vol. 137, pp. 625–645.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kong, L.B. and Jiang, D.Q., Multiple Solutions of a Nonlinear Fourth-Order Periodic Boundary Value Problem, Ann. Polon. Math., 1998, vol. LXIV.3, pp. 265–270.Google Scholar
  15. 15.
    Zhang, M.R., Periodic Solutions of Lienard Equations with Singular Forces of Repulsive Type, J. Math. An. Appl., 1996, vol. 203, no. 1, pp. 254–269.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, M.R., A Relationship between the Periodic and the DirichletBVPs of Singular Differential Equations, Proc. R. Soc. Edinb., Sect. A, 1998, vol. 128, no. 5, pp. 1099–1114.CrossRefzbMATHGoogle Scholar
  17. 17.
    Conti, M., Terracini, S., and Verzini, G., Infinitely Many Solutions to Fourth-Order Superlinear Periodic Problems, Trans. Am. Math. Soc., 2003, vol. 356, pp. 3283–3300.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bouteraa, N. and Benaicha, S., Triple Positive Solutions of Higher-Order Nonlinear Boundary Value Problems, J.Comput. Sci. Comp.Math., 2017, vol. 7, no. 2, pp. 25–31.CrossRefzbMATHGoogle Scholar
  19. 19.
    Torres, P.J., Weak Singularities May Help Periodic Solutions to Exist, J. Diff. Eq., 2007, vol. 232, no. 1, pp. 277–284.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Taliaferro, S., A Nonlinear Singular Boundary Value Problem, Nonlin. An. TMA, 1979, vol. 3, pp. 897–904.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mosconi, S. and Santra, S., On the Existence and Non-Existence of Bounded Solutions for a Fourth-Order ODE, J. Diff. Eq., 2013, vol. 255, no. 11, pp. 4149–4168.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Benaicha, S. and Haddouchi, F., Positive Solutions of a Nonlinear Fourth-Order Integral Boundary Value Problem, Ann. West Univ. Timisoara—Math. Comp. Sci., 2016, vol. 54, no. 1, pp. 73–86.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ding, T.R., A Boundary Value Problem for the Periodic Brillouin Focusing System, Acta Sci. Natur. Univ. Pekinensis, 1965, vol. 11, pp. 31–38.Google Scholar
  24. 24.
    Agarwal, R.P., O’Regan, D., and Wong, P.J.Y., Positive Solutions of Differential, Difference and Integral Equations, Dordrecht: Kluwer, 1999.CrossRefzbMATHGoogle Scholar
  25. 25.
    Ma, R. and Dai, G.W., Periodic Solutions of Nonlocal Semilinear Fourth-Order Differential Equations, Nonlin. An. TMA, A, 2011, vol. 74, no. 15, pp. 5023–5029.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lakshmikantham, V. and Leela, S., Existence and Monotone Method for Periodic Solutions of First-Order Differential Equations, J. Math. An. Appl., 1983, vol. 91, pp. 237–243.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li, Y.X., Positive Solutions of Fourth-Order Periodic Boundary Value Problems, Nonlin. An. TMA, 2003, vol. 54, no. 6, pp. 1069–1078.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Li, W. and Zhang, M.R., Non-Degeneracy and Uniqueness of Periodic Solutions for Some Superlinear Beam Equations, Appl.Math. Lett., 2009, vol. 22, pp. 314–319.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.University of Oran1Ahmed BenbellaAlgeria

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